Introduction BRST global symmetry Anti-field formalism Extensions to 2nd order and current work Cohomological methods for quantum gravity Alex Mitchell SHEP group University of Southampton Southampton student conference 14/5/2019 Alex Mitchell Cohomological methods for quantum gravity
Introduction BRST global symmetry Anti-field formalism Extensions to 2nd order and current work Outline Introduction 1 The tower operator BRST global symmetry 2 The BRST symmetry Nilpotent operators and the BRST cohomology Anti-field formalism 3 Anti-field formalism Relationship to BRST operators The anti-field cascade Extensions to 2nd order and current work 4 Extending to 2nd order Current work Conclusion Alex Mitchell Cohomological methods for quantum gravity
Introduction BRST global symmetry The tower operator Anti-field formalism Extensions to 2nd order and current work Outline Introduction 1 The tower operator BRST global symmetry 2 The BRST symmetry Nilpotent operators and the BRST cohomology Anti-field formalism 3 Anti-field formalism Relationship to BRST operators The anti-field cascade Extensions to 2nd order and current work 4 Extending to 2nd order Current work Conclusion Alex Mitchell Cohomological methods for quantum gravity
Introduction BRST global symmetry The tower operator Anti-field formalism Extensions to 2nd order and current work Based on ArXiv:1802.04281 and ArXiv:1806.02206 by Tim Morris Gravity is difficult to quantize From the renormalization group perspective the coupling 1 2 is irrelevent, i.e. [ κ ] = − 1 κ ∝ G As a result no naive UV complete theory of quantum gravity is possible, despite tricks at low loop level calculations in free gravity Alex Mitchell Cohomological methods for quantum gravity
Introduction BRST global symmetry The tower operator Anti-field formalism Extensions to 2nd order and current work Expanding the metric as H µν = h µν + 1 g µν = δ µν + κ H µν with 2 δ µν φ we find 2 ( ∂ λ h µν ) 2 − 1 L = 1 2 ( ∂ λ φ ) 2 This significantly restricts what eigenoperators can be constructed around the UV Gaussian fixed point, in particular the sign change defines our Sturm-Liouville measure exp ( − ( h µν ) 2 exp ( φ 2 ) 2 Ω Λ ) 2 Ω Λ Alex Mitchell Cohomological methods for quantum gravity
Introduction BRST global symmetry The tower operator Anti-field formalism Extensions to 2nd order and current work This leads to 2 quantization condition equations, where we demand square integrability − ( h µν ) 2 � ∞ O n ( h µν ) O m ( h µν ) = K δ nm −∞ dh µν e 2 ΩΛ φ 2 � ∞ 2 ΩΛ O n ( φ ) O m ( φ ) = K δ nm −∞ d φ e In the second we are no longer permitted polynomials, instead we find our super-relevant tower operators Alex Mitchell Cohomological methods for quantum gravity
Introduction BRST global symmetry The tower operator Anti-field formalism Extensions to 2nd order and current work These tower operators are ( − φ 2 δ ( n ) ∂φ n δ ( 0 ) δ ( 0 ) 2 ΩΛ ) ∂ n 1 Λ ( φ ) := , := √ 2 π Ω Λ e Λ Λ These are perturbative in their couplings, non-perturbative in � and effervescent, an infinite tower of these can be associated to an operator and so these are summed into a ‘coupling function’ f σ c , b , Φ ∗ ) Λ ( φ ) σ ( ∂, ∂φ, h , c , ¯ with n δ ( n ) f σ Λ ( φ ) = � ∞ n = n σ g σ Λ ( φ ) Alex Mitchell Cohomological methods for quantum gravity
Introduction BRST global symmetry The tower operator Anti-field formalism Extensions to 2nd order and current work Unfortunately now we can renormalize every possible interaction we could ever want! This isn’t predictive and we also have an infinite number of couplings However this hasn’t been gravity, we still need to incorporate diffeomorphism invariance to make this a theory of quantum gravity Alex Mitchell Cohomological methods for quantum gravity
Introduction BRST global symmetry The BRST symmetry Anti-field formalism Nilpotent operators and the BRST cohomology Extensions to 2nd order and current work Outline Introduction 1 The tower operator BRST global symmetry 2 The BRST symmetry Nilpotent operators and the BRST cohomology Anti-field formalism 3 Anti-field formalism Relationship to BRST operators The anti-field cascade Extensions to 2nd order and current work 4 Extending to 2nd order Current work Conclusion Alex Mitchell Cohomological methods for quantum gravity
Introduction BRST global symmetry The BRST symmetry Anti-field formalism Nilpotent operators and the BRST cohomology Extensions to 2nd order and current work Before we begin to incorporate diffeomorphism we remind ourselves of Faddeev-Popov ghosts introduced in Yang-Mills theories to gauge fix gauge fields and define their propagator c a + gf abc ( ∂ λ ¯ L ghost = ∂ λ c a ∂ λ ¯ c a ) A b λ c c Famously odd; scalar fields with fermionic statistics and non-asymptotic states, regarded as a useful tool Alex Mitchell Cohomological methods for quantum gravity
Introduction BRST global symmetry The BRST symmetry Anti-field formalism Nilpotent operators and the BRST cohomology Extensions to 2nd order and current work The implementation and expansion of ghosts was expanded upon by Becchi, Rouet, Stora, Tyutin (BRST) who regarded the ghost as a useful field for their global symmetry They introduce a nilpotent operator; an action invariant under the action of this operator is invariant under the gauge symmetry associated to that operator The auxiliary field b µ is also introduced to implement the BRST symmetry off-shell, it has no kinetic term and doesn’t propagate by itself, we also split our fields into ‘families’ Alex Mitchell Cohomological methods for quantum gravity
Introduction BRST global symmetry The BRST symmetry Anti-field formalism Nilpotent operators and the BRST cohomology Extensions to 2nd order and current work ONE TAKE AWAY Diffeomorphism invariance is implemented at the quantum level by requiring a global BRST symmetry (this can be found order by order using the anti-field formalism) Alex Mitchell Cohomological methods for quantum gravity
Introduction BRST global symmetry The BRST symmetry Anti-field formalism Nilpotent operators and the BRST cohomology Extensions to 2nd order and current work We have our BRST operator Q such that QS = 0 It is ’nilpotent’ such that Q 2 O = 0 or more concisely Q 2 = 0 We say that O is ’closed’ under Q e.g. for gravitons and ghosts at the free level Q 0 H µν = ∂ µ c ν + ∂ ν c µ Q 0 c µ = 0 Q 2 0 H µν = Q 0 ( ∂ µ c ν + ∂ ν c µ ) = 0 Alex Mitchell Cohomological methods for quantum gravity
Introduction BRST global symmetry The BRST symmetry Anti-field formalism Nilpotent operators and the BRST cohomology Extensions to 2nd order and current work An operator is exact if it can be defined as O = QK , such that Q O = Q 2 K = 0 Only interested in operators that are closed but not exact. These are in the ‘cohomology’ of the BRST operator We’re free to add exact operators to those in the cohomology, this does not affect the physics and will have important implications for diffeomorphism invariance Alex Mitchell Cohomological methods for quantum gravity
Introduction Anti-field formalism BRST global symmetry Relationship to BRST operators Anti-field formalism The anti-field cascade Extensions to 2nd order and current work Outline Introduction 1 The tower operator BRST global symmetry 2 The BRST symmetry Nilpotent operators and the BRST cohomology Anti-field formalism 3 Anti-field formalism Relationship to BRST operators The anti-field cascade Extensions to 2nd order and current work 4 Extending to 2nd order Current work Conclusion Alex Mitchell Cohomological methods for quantum gravity
Introduction Anti-field formalism BRST global symmetry Relationship to BRST operators Anti-field formalism The anti-field cascade Extensions to 2nd order and current work We can further extend this treatment past Yang-Mills theories to gravity using ‘Batalin-Vilkovisky anti-field formalism’ (or BV or anti-field formalism for short) This is necessary for gravity as here we have much more freedom for field redefinitions; BV accounts for this as well as implementation of BRST symmetry and also now renormalization group flow 3 major parts; the introduction of ‘anti-fields’, the anti-bracket and a measure term Alex Mitchell Cohomological methods for quantum gravity
Introduction Anti-field formalism BRST global symmetry Relationship to BRST operators Anti-field formalism The anti-field cascade Extensions to 2nd order and current work For each field Φ A we associate an anti-field Φ ∗ A with opposite Grassman grading Anti-fields φ ∗ A are introduced, at 1st order, as a source for our BRST transformations S → S + ( Q Φ A )Φ ∗ A Objects conjugate to the ghost anti-field c ∗ µ are the commutator of gauge transformations Free to switch between gauge invariant and gauge fixed actions, results are equivalent. In the former we may use the minimal basis where we can exclude terms involving ¯ c µ and b µ past the free level Alex Mitchell Cohomological methods for quantum gravity
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